Almost Optimal (on the average) Combinatorial Algorithms for Boolean Matrix Product Witnesses, Computing the Diameter (Extended Abstract)

Abstract

We describe almost optimal (on the average) combinatorial algorithms for the following algorithmic problems: (i) computing the boolean matrix product, (ii) finding witnesses for boolean matrix multiplication and (iii) computing the diameter and all-pairs-shortest-paths of a given (unweighted) graph/digraph. For each of these problems, we assume that the input instances are drawn from suitable distributions. A random boolean matrix (graph/digraph) is one in which each entry (edge/arc) is set to 1 or 0 (included) independently with probability p. Even though fast algorithms have been proposed earlier, they are based on algebraic approaches which are complex and difficult to implement. Our algorithms are purely combinatorial in nature and are much simpler and easier to implement. They are based on a simple combinatorial approach to multiply boolean matrices. Using this approach, we design fast algorithms for (a) computing product and witnesses when A and B both are random boolean matrices or when A is random and B is arbitrary but fixed (or vice versa) and (b) computing diameter, distances and shortest paths between all pairs in the given random graph/digraph. Our algorithms run in O(n ²(log n)) time with O(n ^-3) failure probability thereby yielding algorithms with expected running times within the same bounds. Our algorithms work for all values of p.